Fast calculation of stddev

stddev-fast

Datum: x
Size: n
Mean: m

variance = sum((x-m)^2)/n

= (x0-m)^2 + (x1-m)^2 + (x2-m)^2 ...
  ----------------------------------
           n

= x0^2 -2mx0 + m^2 + x1^2 -2mx1 + m^2 + x2^2 -2mx2 +m^2 ...
  ---------------------------------------------------------
           n

= x0^2 + x1^2 + x2^2 + ... + n(m^2) - 2m (x0 + x1 + x2 + ...)
  -----------------------------------------------------------
           n

= sum(x^2) + n(m^2) - 2m sum(x)
  -----------------------------
           n

= sum(x^2) + n (1/n * sum(x)) (1/n * sum(x)) - 2/n sum(x) sum(x)
  --------------------------------------------------------------
           n

= sum(x^2) + 1/n * sum(x)^2 - 2/n sum(x)^2
  ----------------------------------------
           n

= sum(x^2) - 1/n * sum(x)^2
  -------------------------
           n

= n * sum(x^2) - sum(x)^2
  -----------------------
           n^2

stddev = sqrt(variance) = sqrt(n * sum(x^2) - sum(x)^2)/n